I had a similar problem with my T-style Magsafe from my 2,1 MBP.
Here's my odyssey. For the TL;DR-version, buy a new adapter.
I began to notice that my charger would sometimes not work. One minute the light would be a happy green or amber, and the next minute, it would blink out. Sometimes, I could get the charger to start charging again, provided I held the end of the Magsafe connector in "just-so" a position (think adjusting the rabbit ears on an old HF/VHF TV). But once I moved about, it would stop charging again, sometimes. Then I'd have to jiggle with the connector again, hoping that it would start charging. Very intermittent and frustrating.
I tried all the standard remedies. I reset the SMC. I reset the PRAM. I downloaded and installed every appropriate EFI update. I checked to see that the little spring-loaded pins in the Magsafe were clean and had free action. Still nothing.
I remembered that the charging problem started after I'd upgraded to Snow Leopard. So I went back to Tiger, thinking that perhaps it was a software problem. Nope. Back to the drawing board.
I considered the possibility that something was wrong with the battery, even though istat was reporting it as healthy. But trying to run the machine on AC power only (by removing the battery) didn't work at all. The system just wasn't drawing power, whether the battery was attached or not.
I considered the possibility that the left I/O board had failed. To test, I went to the local Best Buy and tried charging my MBP with an L-style Magsafe at the Apple display setup. It started to charge immediately, with no problems whatsoever. I decided that the problem was with my adapter.
I disassembled the Magsafe connector, and found the problem. The spring-loaded pins in the Magsafe are soldered to a tiny little circuit board, which I assume has some logic built-in to properly charge the lithium-ion-based battery chemistry. In my case, the solder joints had broken completely clean off. In its disassembled state (i.e., with the white plastic Magsafe shield removed), however, I could ensure that the pins were properly contacting the circuit board, and lo and behold, my MBP started charging like there never was a problem. Jiggle the connector just enough to cause the pins to lose contact with the circuit board, however, and it stopped charging; this totally explained the weird charging behavior I'd seen prior.
The solder joints were way too tiny for someone of my limited electronics skill to repair. I formulated a different plan. I found an orientation in which the natural lie of the pins (i.e., when gravity was acting upon the connector assembly), when attached to the circuit board, were making correct contact. I then used 2-part epoxy to glue the pins to the circuit board in this orientation; 2-part epoxy is electrically non-conductive. Once the epoxy cured, the connector worked just fine; I'm typing this little novel from my MBP right now.
For the OP, I would suggest using a different adapter. Try going to an Apple Store or Best Buy or similar. If this other adapter works, then you can immediately rule out more complicated problems, like battery or logic board. You'll probably want to get a new adapter and/or perform some surgery on your old one. Since you have a newer L-style Magsafe, the problem may lie in the strain relief, as opposed to the pin connectors; this is a known design flaw that Apple has ignored for years. Just check out the reviews of the L-style Magsafe on Apple's own website. There is also the possibility that your adapter has failed at the power-brick end; the strain relief is also weak on this side as well.
If the other adapter doesn't work either, then you probably have a more serious problem.
And I'd be somewhat wary as to what the Apple "geniuses" tell you. Most of them will immediately accuse the battery, so that they can charge you to open up the machine and tinker with the internals. Caveat emptor.
